Minseong Kwon, Spherical Geometry of Hilbert Schemes of Conics in Adjoint Varieties

For each rational homogeneous space, the space of lines is now well-understood and can be described in terms of the induced group action. It is natural to consider rational curves of higher degree, and in this talk, we discuss geometry of conics in adjoint varieties, which are rational homogeneous spaces associated to simple Lie algebras. Each adjoint variety is equipped with a hyperplane distribution called the contact distribution, and we show that smooth conics transverse to the contact distribution form a homogeneous symmetric variety if the adjoint variety is of Picard number 1. This enables us to view the Hilbert scheme of conics as a spherical variety, and we compute its colored fan by using the description of the space of lines and the Hilbert-Chow morphism.